Question: $f(x) = \sqrt{ 1 - \lvert x \rvert }$ What is the domain of the real-valued function $f(x)$ ?
Solution: $f(x)$ is undefined when the radicand (the expression under the radical) is less than zero. So we know that $1 - \lvert x \rvert \geq 0$ So $\lvert x \rvert \leq 1$ This means $x \leq 1$ and $x \geq -1$ ; or, equivalently, $-1 \leq x \leq 1$ Expressing this mathematically, the domain is $\{ \, x \in \RR \mid -1\leq x \leq1\, \}$.